Updated spacing between @ latex expressions

astrosander edited Oct 6, 2024, 12:48 AM

View solution History

revision #10353 parent #10069 GitHub c702ebe ← older newer →

en/2.1.36.md +1 −1

			<!DOCTYPE html>
			<html lang="en">

			<head>
			<meta charset="utf-8">
			<meta name="viewport" content="width=device-width, initial-scale=1.0">
			<meta http-equiv="content-language" content="en">
			<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
			<meta name="description" content="The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?</title>
			<link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css">
			<link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png">
			<script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script>
			<script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script>
			<script type="text/x-mathjax-config">
			MathJax.Hub.Config({
			extensions: ['tex2jax.js'],
			jax: ['input/TeX', 'output/HTML-CSS'],
			tex2jax: {
			inlineMath: [['$', '$'], ['$', '$']],
			processEscapes: true,
			processClass: 'tex2jax',
			ignoreClass: 'html'
			},
			showProcessingMessages: false,
			messageStyle: 'none'
			});
			</script>
			</head>
			<body style="">
			<header style="text-align:center;">
			<a href="../../" style="text-decoration: none;">
			<div id="logo">
			<span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>
			</div>
			</a>
			<p class="author">
			Solutions of Savchenko Problems in Physics <br>
			<i><b>knowledge must be free</b></i>
			</p>
			</header>

			<h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$2.1.36.$ The velocity of a body of mass $m$ in a viscous liquid decreases with the distance $l$ traveled according to the law $v = v_0 - \beta l$, where $v_0$ is the initial velocity, and $\beta$ is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?
			</p>

			<h3>Solution</h3>
			<p>
			The equation of Newton's second law for the direction of motion:
			$$ ma = F_с $$

			$$ \frac{F_с}{m}=\frac{dv_x}{dt} $$
		@@ -62,7 +62,7 @@
			Derivative of velocity with respect to time
			$$ \frac{dv_x}{dt}=\frac{d}{dt}(v_0-\beta x) $$

		−	$$ \frac{dv_x}{dt}=\frac{dv_0}{dt}-\beta \frac{dx}{dt}=-\beta t $$
		+	$$ \frac{dv_x}{dt}=\frac{dv_0}{dt}-\beta\frac{dx}{dt}=-\beta t $$
			the minus sign shows that the acceleration vector is directed in the direction opposite to the velocity vector.
			Combining the equations, we obtain the value of the resistance force as a function of velocity
			$$ \boxed{F=\beta mv} $$

			</p>

			<h4>Answer</h4>
			<p>
			$$F = \beta mv$$
			</p>


			<footer class="row container">
			<br>
			<p>
			<small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
			</p>
			<p>
			<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
			</p>
			</footer>
			</body>

			</html>
		unchanged lines 21

	<!DOCTYPE html>		<!DOCTYPE html>
	<html lang="en">		<html lang="en">

	<head>		<head>
	<meta charset="utf-8">		<meta charset="utf-8">
	<meta name="viewport" content="width=device-width, initial-scale=1.0">		<meta name="viewport" content="width=device-width, initial-scale=1.0">
	<meta http-equiv="content-language" content="en">		<meta http-equiv="content-language" content="en">
	<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">		<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
	<meta name="description" content="The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?">		<meta name="description" content="The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?">
	<meta name="author" content="Aliaksandr Melnichenka">		<meta name="author" content="Aliaksandr Melnichenka">
	<meta name="date" content="2023-10" scheme="YYYY-MM">		<meta name="date" content="2023-10" scheme="YYYY-MM">
	<meta property="og:title" content="The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?">		<meta property="og:title" content="The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?">
	<meta property="og:image" content="img/logo.png">		<meta property="og:image" content="img/logo.png">
	<meta property="og:description" content="The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?">		<meta property="og:description" content="The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?">
	<meta name="yandex-verification" content="6cfda41f74038368">		<meta name="yandex-verification" content="6cfda41f74038368">
	<title>The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?</title>		<title>The velocity of a body of mass m in a viscous liquid decreases with the distance l traveled according to the law v = v_0 - \beta l, where v_0 is the initial velocity, and \beta is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?</title>
	<link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css">		<link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css">
	<link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png">		<link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png">
	<script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script>		<script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script>
	<script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script>		<script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script>
	<script type="text/x-mathjax-config">		<script type="text/x-mathjax-config">
	MathJax.Hub.Config({		MathJax.Hub.Config({
	extensions: ['tex2jax.js'],		extensions: ['tex2jax.js'],
	jax: ['input/TeX', 'output/HTML-CSS'],		jax: ['input/TeX', 'output/HTML-CSS'],
	tex2jax: {		tex2jax: {
	inlineMath: [['$', '$'], ['$', '$']],		inlineMath: [['$', '$'], ['$', '$']],
	processEscapes: true,		processEscapes: true,
	processClass: 'tex2jax',		processClass: 'tex2jax',
	ignoreClass: 'html'		ignoreClass: 'html'
	},		},
	showProcessingMessages: false,		showProcessingMessages: false,
	messageStyle: 'none'		messageStyle: 'none'
	});		});
	</script>		</script>
	</head>		</head>
	<body style="">		<body style="">
	<header style="text-align:center;">		<header style="text-align:center;">
	<a href="../../" style="text-decoration: none;">		<a href="../../" style="text-decoration: none;">
	<div id="logo">		<div id="logo">
	<span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>		<span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>
	</div>		</div>
	</a>		</a>
	<p class="author">		<p class="author">
	Solutions of Savchenko Problems in Physics <br>		Solutions of Savchenko Problems in Physics <br>
	<i><b>knowledge must be free</b></i>		<i><b>knowledge must be free</b></i>
	</p>		</p>
	</header>		</header>

	<h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3>		<h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3>

	<h3> Statement </h3>		<h3> Statement </h3>
	<p>		<p>
	$2.1.36.$ The velocity of a body of mass $m$ in a viscous liquid decreases with the distance $l$ traveled according to the law $v = v_0 - \beta l$, where $v_0$ is the initial velocity, and $\beta$ is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?		$2.1.36.$ The velocity of a body of mass $m$ in a viscous liquid decreases with the distance $l$ traveled according to the law $v = v_0 - \beta l$, where $v_0$ is the initial velocity, and $\beta$ is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body?
	</p>		</p>

	<h3>Solution</h3>		<h3>Solution</h3>
	<p>		<p>
	The equation of Newton's second law for the direction of motion:		The equation of Newton's second law for the direction of motion:
	$$ ma = F_с $$		$$ ma = F_с $$

	$$ \frac{F_с}{m}=\frac{dv_x}{dt} $$		$$ \frac{F_с}{m}=\frac{dv_x}{dt} $$
	@@ -62,7 +62,7 @@
	Derivative of velocity with respect to time		Derivative of velocity with respect to time
	$$ \frac{dv_x}{dt}=\frac{d}{dt}(v_0-\beta x) $$		$$ \frac{dv_x}{dt}=\frac{d}{dt}(v_0-\beta x) $$

	$$ \frac{dv_x}{dt}=\frac{dv_0}{dt}-\beta \frac{dx}{dt}=-\beta t $$		$$ \frac{dv_x}{dt}=\frac{dv_0}{dt}-\beta\frac{dx}{dt}=-\beta t $$
	the minus sign shows that the acceleration vector is directed in the direction opposite to the velocity vector.		the minus sign shows that the acceleration vector is directed in the direction opposite to the velocity vector.
	Combining the equations, we obtain the value of the resistance force as a function of velocity		Combining the equations, we obtain the value of the resistance force as a function of velocity
	$$ \boxed{F=\beta mv} $$		$$ \boxed{F=\beta mv} $$

	</p>		</p>

	<h4>Answer</h4>		<h4>Answer</h4>
	<p>		<p>
	$$F = \beta mv$$		$$F = \beta mv$$
	</p>		</p>


	<footer class="row container">		<footer class="row container">
	<br>		<br>
	<p>		<p>
	<small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>		<small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
	</p>		</p>
	<p>		<p>
	<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>		<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
	</p>		</p>
	</footer>		</footer>
	</body>		</body>

	</html>		</html>
	unchanged lines 21